On the other side of the case are actual Borgheses: Princess Marcella’s son Francesco; his wife, Amanda; and his two sons Scipione and Lorenzo, the latter probably best known as the charming prince on “The Bachelor,” the ABC reality show. The Borgheses have a beauty line, as well, which they sell on the Home Shopping Network, and Lorenzo has a line of products for pets called “The Royal Treatment.”

None of the products are sold under the family name, but their marketing does play up the Borgheses’ noble lineage. Borghese Inc. says that the heritage isn’t theirs to capitalize on anymore, and is suing the Borgheses and demanding that they stop referring to their family history or drawing any links to it during the promotion or sale of any products.

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The Borghese family, Amanda, Lorenzo and Francesco. A lawsuit over the use of the family history will begin trial this summer.Credit
Nadav Neuhaus for The New York Times

“This is no different than if any other brand name with a surname like Lauder, McDonald, Heinz, Gallo, Ferragamo were to take steps as they do to stop others from using their intellectual property rights,” said Mark N. Mutterperl, the lawyer representing Borghese Inc. “Our client is not a company that runs around suing people every day.” (Ms. Mosbacher and the company declined to comment.)

The lawsuit, which was first filed in July 2010 and faces pretrial deadlines on Friday, has already proved so costly that Judge J. Paul Oetken, the Federal District Court judge handling the case, compared it to the Jarndyce v. Jarndyce lawsuit in Charles Dickens’s “Bleak House.” In an April 29 ruling, he said he would not offer further extensions and set plans for a trial in July “or August at the latest.”

Mark Evens, the lawyer representing the Borghese descendants, agrees that the case should be expedited.

“It has gone too long,” Mr. Evens said. The company suing the family “has suffered no harm. No dilution of their mark. No counterfeiting. No palming off. That’s the tragedy here.”

And Francesco Borghese, Princess Marcella’s son, says he is especially troubled by being sued since he said he had essentially retired from his own cosmetics company a decade ago. “The last thing I needed was a lawsuit at the tender age of 72,” said Mr. Borghese, who will turn 75 in December. “I was just trying to build a business and give something to my children.”

Lorenzo Borghese has been the most visible member of the family. Since his appearance on “The Bachelor” in 2006, he has been spotted with New York socialites like Georgina Bloomberg, Mayor Michael R. Bloomberg’s daughter, and Tinsley Mortimer. And his historical novel “The Princess of Nowhere,” about the stormy marriage of one of his ancestors to Napoleon’s sister, Pauline Bonaparte Borghese, was published in 2010.

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Georgette MosbacherCredit
Rick Wilking/Reuters

“What are they going to get from us? They are not going to get our history,” Lorenzo Borghese said over lunch at Bottega del Vino in Midtown Manhattan. “They believe they own my family’s history for everything.”

Kenneth L. Port, a law professor and director of the Intellectual Property Institute at the William Mitchell College of Law in St. Paul, Minn., said there were a growing number of disputes like this in the courts.

He cited the dispute between North Face and Jimmy Winkelmann, a 16-year-old who started a clothing line called South Butt and North Face. Chick-fil-A sued a Vermont folk artist over a trademark to the phrase “Eat More Kale,” which the company argues overlaps with its slogan “Eat More Chicken.”

“We’re seeing a growth because trademark owners are finding that the more kind of bullying conduct they do, the more the trademark is worth,” Mr. Port said. “They think they have to act like a bully to get the trademark stronger.”

But David S. Welkowitz, a law professor at Whittier Law School in Costa Mesa, Calif., said the Borghese case could not be considered bullying because “the Borghese family sold their rights to the name.”

For decades, the Borghese family and the Borghese cosmetics business coexisted with little dispute.

In 1976, Revlon bought the rights, title and interest to the Borghese cosmetics brand, including what court papers said were “the words and phrases BORGHESE, MARCELLA BORGHESE, PRINCESS MARCELLA BORGHESE.” In 1992, Revlon sold the Borghese company and Ms. Mosbacher became its chief executive and soon reached an agreement with the family for final payments, which also is a matter of dispute.

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Princess Marcella Borghese at the Galleria Borghese in Rome, circa 1992, with a statue of Pauline Bonaparte Borghese.

Over the years, Princess Marcella’s descendants carved out their own niches in the beauty business.

Francesco started his own line of beauty products in the early 1980s in the United States under names like Orlane, Perlier and Elariia, and starting in the 1990s, the family began appearing on the home shopping channel QVC and then, later, on HSN.

Relations with Borghese Inc., which is privately held, started to sour in 2006 when Lorenzo Borghese started working with ABC about possibly appearing on “The Bachelor.” Mr. Borghese contacted the company about serving as a consultant. The television program filmed a scene in the offices of Borghese Inc.

But after ABC issued a news release in 2006 that mentioned Lorenzo’s grandmother and said she had “started the famed self-named cosmetics line, Borghese Inc.,” Ms. Mosbacher wrote to Mr. Borghese warning him against “causing any false impression in the marketplace that there is a connection or relationship between yourself and Borghese Inc. and our cosmetics products.”

In 2008, the companies came to blows again when Lorenzo applied for a federal trademark for a line of pet shampoos and conditioners called “La Dolce Vita by Prince Lorenzo Borghese” for PetSmart. Borghese Inc. contested the trademark. As the trademark neared approval in 2010, Borghese Inc. sued.

The Borgheses say they are reaching their financial limits and have paid $4 million in legal fees. But Lorenzo Borghese says the case is worth pursuing for the family name.

“They don’t even want us to mention that we are the family,” he said.

A version of this article appears in print on June 16, 2013, on page A1 of the New York edition with the headline: Borghese v. Borghese: Battle for a Royal Name. Order Reprints|Today's Paper|Subscribe